SOLUTION: what is the equation of circle which touches x axis at (4,0) and cuts up an intercept 8 units from the positive y axis?

Question 890591: what is the equation of circle which touches x axis at (4,0) and cuts up an intercept 8 units from the positive y axis?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Something in your problem is unclear or was left out, so we cannot solve it without understanding what was meant by the problem.
I will make my best guess at what the problem meant, and I will solve that guessed problem. (No guarantees).

In general, you need 3 points to determine a circle,
so if we knew 2 different points where the circle crosses the x-axis,
we would only need another (third) point.
However, "touches x axis at (4,0)"
gives us point P(4,0) and strongly suggests the extra information that the x-axis is tangent to the circle
(that the circle touches the x-axis only at that point),
so that information is worth the same as 2 different points.
What I can only guess is what is meant by saying that
the circle "cuts up an intercept 8 units from the positive y axis"

The radius to the point of where a circle is tangent to a line is perpendicular to the line,
so we know that the radius is the vertical line .
From that we have that the center is at , where is the radius,
but we do not know yet:

Maybe what was meant is that the circle cuts a segment of length 8,
between its two intercepts, out of the positive y-axis, like this:
and the blue isosceles triangle is key to find .
Drawing a horizontal line to the center of the circle, I split into two right triangles ("coincidentally" also isosceles)
Then we apply the Pythagorean theorem to get .
From there we get to ---> ---> --->
and then, knowing the coordinates of the center and the radius, we can write the equation of the circle:

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